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For a class of generalized holomorphic Eisenstein series, we establishcomplete asymptotic expansions (Theorems~1~and~2), which together with theexplicit expression of the latter remainder (Theorem~3), naturally transfer toseveral new variants of the celebrated formulae of Euler and of Ramanujan forspecific values of the Riemann zeta-function (Theorem~4 andCorollaries~4.1--4.5), and to various modular type relations for the classicalEisenstein series of any even integer weight (Corollary~4.6) as well as forWeierstra{\ss}' elliptic and allied functions (Corollaries~4.7--4.9). Crucialr{\^o}les in the proofs are played by certain Mellin-Barnes type integrals,which are manipulated with several properties of confluent hypergeometricfunctions.

Asymptotic expansions for a class of generalized holomorphic Eisenstein  series, Ramanujan's formula for $\zeta(2k+1)$, Weierstrass' elliptic and  allied functions